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CHAPTER 1

Introduction

Mathematical sociology is not an oxymoron. There is a useful role formathematics in the study of society and groups. In fact, that role is growingas social scientists and others develop better and better tools for the studyof complex systems. A number of trends are converging to make theapplication of mathematics to society increasingly productive.

First, more and more human systems are complex, in a sense to bedescribed soon. World economies are more and more interconnected. Ourtransportation and communication systems are increasingly worldwide.Social networks are less local and more global, making them morecomplex, producing new emergent communication patterns, a positiveeffect, but which also has made us increasingly vulnerable to pandemics,a negative effect. The Internet has connected us in ways that no oneunderstands completely. Power grids are less and less local and aresubject to more widespread failures than ever before. New species areincreasingly introduced into local complex ecologies with unexpectedeffects. Our recent climate change has produced a situation in which itis more and more important to predict the future and the effects of humaninterventions in the complex system of the global weather. The mapping ofthe human genome makes available to biologists the possibility of studyingthe complex system of interactions between genes and proteins. All ofthese tendencies mean that scientists in a wide variety of areas—computerscience, economics, ecology, genetics, climatology, epidemiology, andothers—have developed mathematical tools to study complex systems, andthese tools are available to us sociologists.

The second important trend is the growing power and ubiquity ofthe computer. Computer simulations and mathematics are complementarytools for the study of complex systems. They are two different ways ofdrawing implications for the future from what is known or assumed tobe true. Mathematics can be used to draw far-reaching and sometimesunexpected conclusions using logic and mathematics. For example, manyproperties of networks have been proved to be true by mathematiciansusing traditional mathematical tools. Computer simulations use computerprograms the coding of which embodies assumptions and whose conclu-sions are evident after the program has iterated. Computer simulations areuseful in situations that are unsolved or intractable mathematically.

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2 • Chapter 1

This text uses both mathematics and computer simulations. Sometimesthe computer simulations demonstrate phenomena for which there is noexact mathematical solution. More frequently simulations are used toillustrate some model so that you, the reader of this book, will gain someunderstanding of how the model works and how it is affected by varyingparameters even if a full mathematical treatment of the model is beyondthe purpose of this book.

EPIDEMICS

At the time we are writing this chapter there is an epidemic of concern overswine fever, a variant of influenza that seems to have captured the public’sattention. Both the flu and fear of this flu spread through social networks,and we want now to illustrate some of the properties of epidemics througha very simple model. The model will be illustrated both with a little simplemathematics and with a computer simulation.

Suppose that a large population consists of N individuals. Supposethat each individual in the population has small probability p of beingconnected to each of the others in the population and that his connectionto one individual has no bearing on his connection to any other personin the population. This creates a random network among the membersof the population. Real social networks are not like random networks,but random networks are very tractable mathematically, and so they tendto be assumed by epidemiologists who study the spread of diseases. Thepowerful conclusions may be relatively unaffected by the unreality of theassumptions, much like the statistician may on occasion assume a normaldistribution because the conclusions are not affected very much if theassumption of a normal distribution is not exactly true. We will examinehow real social networks differ from random networks in much more detailin later chapters, but for the moment we will assume that they are usefuldescriptions of real networks.

Despite its unrealism, let’s, for the sake of convenience, assume thatthe network is random. Suppose that initially just one person is sick witha contagious disease. If p is the probability this person is connected toanyone else, then we should expect this person to be connected to p × Nothers, on the average. If p is small and N is large, then each of thesepN individuals will be connected to pN others, and so the sick individualwill be connected to (pN)2 individuals indirectly, through his contacts.If p is small and N is large we can ignore the unlikely event that someindividuals will be connected to more than one of his direct connections.Similarly (pN)3 persons will be connected even less directly, throughtwo intermediaries, and (pN)k+1 persons will be connected through kintermediaries. Let’s look at this sequence:

1 + pN + (pN)2 + (pN)3 + (pN)4 + · · · (1.1)



Introduction • 3

Figure 1.1. Total number of infected individuals when pN = 1.5.

This represents the expected or average number of people who get thecontagious disease carried by one person. Of course, this number must infact be limited because the population is of finite size N, but if pN ≥ 1,this sum diverges: it just gets bigger and bigger, without limit. If pN < 1,however, the sum converges to a number, and this number may be quitesmall relative to N. The sequence in equation 1.1 converges to,

1 + pN + (pN)2 + (pN)3 + (pN)4 + · · · = 11 − pN

(1.2)

You can verify this for yourself by substituting a few values. WhenpN = .5, for example, the sums are 1, 1.5, 1.75, 1.875, and so on, gettingcloser and closer to 2, the limit. What this means is that if an infectedperson infects, on the average, less than one other person the disease willnot become an epidemic affecting nearly everyone, but otherwise it willspread to the entire population.

The following figures were generated from a simulation based on a fewsimple assumptions. The network was of size 2,500. Ten individuals wereinitially infected. The probability that any two people were connected, alsothe probability that an infected person would infect a healthy person in anygiven time period, was set at .0006: p = .0006. The network was examinedover 100 time periods. If infected in one time period, the person wasassumed to be infectious at the next time period and immune thereafter.In this case p× N = .0006× 2,500 = 1.5, and 1.5 is bigger than the criticalvalue of 1.00 ties per person. We should expect the disease to spread.

Figure 1.1 shows the total number infected. After just 25 time periodsmost had been infected. The curve shows a familiar S-shaped figure, calledthe logistic curve. The disease spread slowly when few had it, then pickedup speed, then slowed down as there were fewer and fewer who had notbeen infected and were not immune. Figure 1.2 shows the number of newcases in each time period, telling the same story as Figure 1.1 in a differentway.



4 • Chapter 1

Figure 1.2. Number of newly infected individuals over time when pN = 1.5.

Figure 1.3. Number of total and newly infected individuals over time whenpN = .75.

Now suppose we make the parameter p half as big, p = .0003, so thateach individual averages .75 contacts instead of 1.50. This is below thecritical value of one contact per person on average. We would expect theoutbreak not to become an epidemic and to peter out after the initial setof infected individuals fail to reproduce themselves. Figure 1.3 shows thatthe total number never passes 50 and illustrates the declining number ofnew cases.

In later chapters we’ll see how these inferences are a consequence ofdeep and nonobvious properties of random networks. We will also see howthe inferences must be modified for other classes of nonrandom networks.Even with these qualifications, the results are interesting. First, they applyto phenomena other than the spread of disease. Information and rumortransition can also be modeled. Coleman et al. (1966) used this modelto account for diffusion in the use of a new antibiotic among doctors inMidwestern communities. In this case, what was being spread was notdisease but information and influence. The existence of a critical pointhas policy implications in epidemiology. It means that not everyone needbe inoculated for an inoculation campaign to be effective. It also helpsexplain why Apple computers are less subject to viruses than are PCs.Since computer viruses are targeted for one operating system, any Mac



Introduction • 5

virus will spread only from Mac to Mac, while almost all the computersa Mac is connected to will be PCs. The Mac to Mac network will be verysparse while the PC to PC network will have a lot higher density (a highervalue of p). Thus, the lower frequency of Mac infections need not be dueto any superiority of the Mac operating system but simply due to the factthat very few people use Macs.

The spread of a disease or information depends not only on thedensity of the network but also on the presence or absence of long-range connections in the network, and this topic can be examined bothmathematically and through the use of computer simulations. When inhistory almost all ties between people were strictly local, epidemic diseasesspread much more slowly. The “Black Death,” a plague that decimatedEurope in the Middle Ages, was carried by sailors from Asia to Italian portcities in 1347, but it did not reach England until 1349 and Scandinavia until1350. This slow spread occurred because long-distance movements hardlyexisted. Most people never saw anyone outside their own small village.Nowadays worldwide influenza pandemics occur every year.

Simulated Epidemics

Let’s explore this difference using the demonstration Simulated Epidemics.This demonstration offers the possibility of examining contagion in twodifferent types of networks, random networks and grids. In grid networksall ties are local, like in a farming community where farmers have relationsonly to those in neighboring farms; there are no long-distance ties. Ina random network there are no constraints on the ties at all. In thedemonstration, the number of individuals and connections are the samein a grid and random network: there are 400 individuals and the averagenumber of connections is four. Play with the demonstration for a whileand you see that there are two major easily observable differences. First,the diseases spread much more rapidly in the random network. Second,the shape of the curves for new cases is quite different. In the grid thenumber of new cases goes up in a straight line until the edges of the gridare reached. In the random network the number of cases seems to go upexponentially at the beginning.

Why are there these differences? In a grid the disease can expand onlyat the circumference of the infected area. It is only individuals at the borderof the infected area who come in contact with uninfected individuals. Onthe other hand, in a random network each newly individual in the earlystages is coming in contact with four uninfected individuals. In a randomnetwork we will have 1 infected individual, then 4, then each of these 4will infect 4 others so that there will be 16 new cases, then each of these16 cases will infect 4 more for 64 more cases, and so on. The number ofnew cases will not be proportional to time, t, but 4t.



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RESIDENTIAL SEGREGATION

Residential segregation by race and class has many causes. In the pastsome of it was legal: residential covenants prevented whites from sellingtheir homes to nonwhites. Patterns established in this way persist. Somesegregation is economic because race and income are correlated and neigh-borhoods with different priced homes may become segregated primarily byincome but also incidentally by race. However, some segregation patternsresult from individual choices: individuals may wish to avoid being aminority in their own community, or at least being a small minority. Suchvoluntary segregation based on the desire to be in proximity to similarothers can produce segregation by sex or social class as well as race. In alunchroom of young children or at a party among middle-aged adults, theremay be segregation by gender. People of different social classes don’t meeteach other in unstructured situations, but our experience in public schoolswas that school social events and cafeteria lunch groups were segregatedby class, sex, and race.

Thomas Schelling (1969, 1972), an economist, was the first to explorethis phenomenon in simulations. Schelling devised what would now becalled an agent-based model. In this simulation individual actors wereplaced in a two-dimensional grid, an eight by eight checkerboard. Actorswere of two different types: “X” and “O” actors. Each actor could changehis position, following certain simple rules. Each actor wanted a minimumpercentage of her neighbors to be of the same type. On this grid each actorhad up to eight neighbors: to the right, left, above, below, upper right,upper left, lower right, and lower left. Actors at the corners of the grid hadonly three neighbors, while actors on an edge but not a corner had fiveneighbors. Some squares of the grid would be empty, to permit movement.

Each actor followed a simple rule: he required that a certain minimumproportion of his neighbors be of his type. Suppose, for example, thatdesired minimum proportion were greater than one-third. Then if a personhad just one neighbor, that actor would have to be of the same type; iftwo neighbors, at least one must be of the same type; if three, four, or fiveneighbors, at least two; and if six, seven, or eight neighbors, at least three.If an actor were dissatisfied she would move to the nearest position on thegrid that was satisfactory in terms of the composition of the neighborhood.

The interesting thing about such situations is that they can lead tocascades of movement. If one actor moves he shifts the character of hisprevious neighborhood, increasing the prominence of the other type, sothat others who were satisfied before his move may, as a consequence ofhis move, become dissatisfied with the composition of their neighborhood.These cascades are emergent properties of the simulation: there is usuallyno simple way of predicting the outcome. Schelling showed that even suchsimple rules could have unexpected outcomes.

Schelling presented this model in 1972, when computers were not asprominent in research as they are today. He used a small eight-by-eight grid



Introduction • 7

Figure 1.4. Initial locations of actors in the Schelling simulation.

that he could manipulate by eye, instead of the much larger grid he couldhave used even with the computers of his day. However, there is anothereffect of his choice not to use computers. Using a checkerboard-sized gridand moving the pieces by eye emphasized his larger point—that simpleassumptions, easily implemented, can produce unexpected outcomes.

Of course, there are outcomes that are not surprising. If neither typeis willing to be a minority in its neighborhood, the result will be completesegregation of the pieces, each type occupying its own separate areas of thegrid. But suppose that each actor wants her own type to constitute morethan a third of her neighbors. Figure 1.4 shows an initial distribution withabout one-third X type, one third O type, and one third empty squares.



8 • Chapter 1

Figure 1.5. Schelling grid after one move.

Squares in which one-third or less of the neighbors are of the same type(the dissatisfied actors) are shown in a smaller font.

In all, 27 actors are satisfied with their neighborhoods and 15 are not.On the average, 45% of anyone’s neighbors are of the same color. Supposethat one of the unhappy actors moves. For example, the X actor in the thirdrow and first column has five neighbors, only one of whom is a fellow X.A move to the square in the second row and third column will give himthree X and three O neighbors, 50% with his type. Figure 1.5 shows thegrid after his move.

As a result of this move, not only is he (the actor initially on (3,1) whomoved to (2,3)) happier, but the actor in the third row and third columnnow also has more than one-third of his neighbors of his type.



Introduction • 9

Figure 1.6. Schelling grid after sixteen moves.

Figure 1.6 shows the results after 16 moves. Note that all actors livein acceptable neighborhoods: no letters are in small fonts. The degree ofsegregation into primarily X and O regions is quite evident. One possiblemeasure of segregation is the average proportion of neighbors who are ofthe same type. Figure 1.7 shows that this measure climbed regularly. Bythe sixteenth move the degree of segregation is almost three-quarters, eventhough the actors are programmed to require that more than one third oftheir neighbors be of their own color. This is one of the surprising aspectsof this simulation—the high degree to which the actual segregation exceedsthe minimum desired degree. This high proportion is clearly an emergentproperty of the simulation. As Schelling points out, no one may want tolive in such a highly segregated society. Everyone may actually prefer more



10 • Chapter 1

Figure 1.7. Segregation after sixteen moves.

integration, but a highly segregated society can emerge from independentactions by unwitting actors who do take the larger context into account.

Schelling’s Model of Residential Segregation

This demonstration enables you to examine Schelling’s model and tovary its parameters. You can change the relative sizes of the majorityand minority (represented by blue and green squares), the proportion ofempty squares, and the preferences of the majority and minority for havingneighbors of their own color. Some homework problems also use thisdemonstration. In this chapter we have tried to give you, the reader andteacher, a sense of what this book is like. No mathematical backgroundbeyond simple algebra is assumed. All additional mathematics will beintroduced as needed. The tools are simple mathematical models andcomputer simulations of social processes. Computer simulations are usedwhen the phenomenon itself is intractable mathematically and to give youthe reader a sense of how a mathematical model develops and is sensitiveto changes in its parameters. The emphasis is on conclusions that wouldnot be obvious without the use of mathematics or simulations. Theseare the unexpected emergent properties of social systems of complexlyinterconnected individuals, and it is the study of these emergent propertiesthat constitutes the bailiwick of sociology.

Chapter Demonstrations

• Schelling’s Model of Residential Segregation simulates the Schellingmodel of residential segregation.



Introduction • 11

• Simulated Epidemics simulates the spread of an epidemic in a randomgrid network. The density of contagious connections in the networkcan be varied.

EXERCISES

1. The simulation Simulated Epidemics allows for variation in the size ofa population living on a square grid (the size of the population is thesquared length of a side of the grid) and the probability of a randomconnection between any two nodes, labeled “Density of Network.”Below the critical value of the density, the disease dies out and abovethat it becomes an epidemic. What are the corresponding critical valuesof the density for each grid? Determine these results theoretically fromwhat the chapter says about critical values.

2. Verify these answers by showing, with Simulated Epidemics, that values50% larger than the critical value of the density produce epidemics thatspread to most of the population.

3. For a 50 by 50 grid, what would you expect to be the effect of networkdensity on the time it takes for the epidemic to reach 80% of thepopulation? Would increasing the density increase or decrease the time ittakes for the epidemic to infect 80% of the population? Provide evidencefor your conjecture by running the simulation with different densities.

4. Using the Schelling demonstration, create a community in which 50% ofthe squares are green, 25% are blue, and both types prefer that at leastone-third of their neighbors are the same type. Move unhappy squares aslong as you can (the simulation will stop when no square is dissatisfiedwith its neighborhood). Do this a few times. The segregation index issimply the average proportion of same-color neighbors. Although peopleare willing to live in neighborhoods with one-third of the other color,what is the average value of the index?

5. Now change the last simulation so that both types of squares want to livein neighborhoods in which their color is in the majority. What happensto the segregation index?


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